Geometry of random planar maps with high degrees

Start:
April 28, 2016, 3:30 p.m.

End:
April 28, 2016, 4:30 p.m.

Speaker: Budd, Timothy (Copenhagen University)
Title:
Geometry of random planar maps with high degrees
Abstract: Many classes of random planar maps, i.e. planar graphs embedded in the sphere modulo homeomorphisms, possess scaling limits (in a Gromov-Hausdorff sense) described by a universal random continuum metric space known as the Brownian map. One way to escape this universality class is to consider certain Boltzmann planar maps with carefully tuned weights on the faces. Le Gall and Miermont have shown that the scaling limits of these fall into a larger class of continuum random metric spaces, often referred to as the stable maps. In this talk I will consider the geometry of the dual (in the planar map sense) of these Boltzmann planar maps, which shows quite different scaling behavior. In particular, I will discuss the growth of the perimeter and volume of certain geodesic balls of increasing radius based on the examination of a peeling process. For a particular range of parameters, in the so-called the dilute phase, precise scaling limits may be obtained for these processes, which suggests possibility of a non-trivial scaling limit in the Gromov-Hausdorff sense. Depending on time I'll discuss a more detailed scaling limit in terms of growth-fragmentation processes and/or the connection with O(n) loop models coupled to random planar maps. Based on joint work with Nicolas Curien, and partially also with Jean Bertoin and Igor Kortchemski.